Abstract
We generalize Minami’s estimate for the Anderson model and its extensions to n eigenvalues, allowing for n arbitrary intervals and arbitrary single-site probability measures with no atoms. As an application, we derive new results about the multiplicity of eigenvalues and Mott’s formula for the ac-conductivity when the single site probability distribution is Hölder continuous.
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Aizenman, M.: Localization at weak disorder: some elementary bounds. Rev. Math. Phys. 6, 1163–1182 (1994)
Aizenman, M., Molchanov, S.: Localization at large disorder and extreme energies: an elementary derivation. Commun. Math. Phys. 157, 245–278 (1993)
Aizenman, M., Schenker, J., Friedrich, R., Hundertmark, D.: Finite volume fractional-moment criteria for Anderson localization. Commun. Math. Phys. 224, 219–253 (2001)
Aizenman, M., Warzel, S.: On the joint distribution of energy levels of random Schrödinger operators. Preprint
Bellissard, J., Hislop, P., Stolz, G.: Correlation estimates in the Anderson model. J. Stat. Phys. 129, 649–662 (2007)
Carmona, R., Klein, A., Martinelli, F.: Anderson localization for Bernoulli and other singular potentials. Commun. Math. Phys. 108, 41–66 (1987)
Combes, J.M., Germinet, F., Klein, A.: Poisson statistics for eigenvalues of continuum random Schrödinger operators. arXiv:0807.0455v3 [math-ph]
Combes, J.M., Hislop, P.D.: Localization for some continuous, random Hamiltonians in d-dimension. J. Funct. Anal. 124, 149–180 (1994)
Combes, J.-M., Hislop, P.D., Klopp, F.: Regularity properties for the density of states of random Schrodinger operators. In: Waves in Periodic and Random Media, South Hadley, MA, 2002. Contemp. Math., vol. 339, pp. 15–24 (2003)
Combes, J.M., Hislop, P.D., Klopp, F.: Optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators. Duke Math. J. 140, 469–498 (2007)
Delyon, F., Lévy, Y., Souillard, B.: Anderson localization for multidimensional systems at large disorder or large energy. Commun. Math. Phys. 100(4), 463–470 (1985)
von Dreifus, H., Klein, A.: A new proof of localization in the Anderson tight binding model. Commun. Math. Phys. 124, 285–299 (1989)
Fröhlich, J., Martinelli, F., Scoppola, E., Spencer, T.: Constructive proof of localization in the Anderson tight binding model. Commun. Math. Phys. 101, 21–46 (1985)
Fröhlich, J., Spencer, T.: Absence of diffusion with Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88, 151–184 (1983)
Germinet, F., Klein, A.: Bootstrap multiscale analysis and localization in random media. Commun. Math. Phys. 222, 415–448 (2001)
Germinet, F., Klein, A.: A characterization of the Anderson metal-insulator transport transition. Duke Math. J. 124, 309–351 (2004)
Germinet, F., Klein, A.: New characterizations of the region of complete localization for random Schrödinger operators. J. Stat. Phys. 122, 73–94 (2006)
Graf, G.-M., Vaghi, A.: A remark on an estimate by Minami. Lett. Math. Phys. 79, 17–22 (2007)
Hundertmark, D.: On the time-dependent approach to Anderson localization. Math. Nachr. 214, 25–38 (2000)
Killip, R., Nakano, F.: Eigenfunction statistics in the localized Anderson model. Ann. H. Poincaré 8, 27–36 (2007)
Kirsch, W.: An invitation to random Schrödinger operators. In: Panorama et Synthèses (2008)
Klein, A., Lenoble, O., Müller, P.: On Mott’s formula for the ac-conductivity in the Anderson model. Ann. Math. 166, 549–577 (2007)
Klein, A., Molchanov, S.: Simplicity of eigenvalues in the Anderson model. J. Stat. Phys. 122, 95–99 (2006)
Kritchevski, E.: Poisson statistics of eigenvalues in the hierarchical Anderson model. Ann. H. Poincaré 9, 685–709 (2008)
Minami, N.: Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Commun. Math. Phys. 177, 709–725 (1996)
Nakano, F.: The repulsion between localization centers in the Anderson model. J. Stat. Phys. 123, 803–810 (2006)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, New York (1980), revised and enlarged edition
Simon, B.: Cyclic vectors in the Anderson model. Special issue dedicated to Elliott H. Lieb. Rev. Math. Phys. 6, 1183–1185 (1994)
Simon, B., Wolff, T.: Singular continuum spectrum under rank one perturbations and localization for random Hamiltonians. Commun. Pure Appl. Math. 39, 75–90 (1986)
Stoiciu, M.: The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle. J. Approx. Theory 139, 29–64 (2006)
Stoiciu, M.: Poisson statistics for eigenvalues: from random Schrödinger operators to random CMV matrices. In: Probability and Mathematical Physics. CRM Proc. Lecture Notes, vol. 42, pp. 465–475. Amer. Math. Soc., Providence (2007)
Wegner, F.: Bounds on the density of states in disordered systems. Z. Phys. B 44, 9–15 (1981)
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A.K was supported in part by NSF Grant DMS-0457474.
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Combes, JM., Germinet, F. & Klein, A. Generalized Eigenvalue-Counting Estimates for the Anderson Model. J Stat Phys 135, 201–216 (2009). https://doi.org/10.1007/s10955-009-9731-3
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DOI: https://doi.org/10.1007/s10955-009-9731-3